Properties

Label 2-77-77.19-c1-0-0
Degree $2$
Conductor $77$
Sign $0.380 - 0.924i$
Analytic cond. $0.614848$
Root an. cond. $0.784122$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.30 + 0.136i)2-s + (2.02 + 1.82i)3-s + (−0.277 + 0.0590i)4-s + (0.166 + 0.373i)5-s + (−2.89 − 2.10i)6-s + (−2.50 + 0.847i)7-s + (2.84 − 0.924i)8-s + (0.466 + 4.43i)9-s + (−0.268 − 0.464i)10-s + (3.26 − 0.590i)11-s + (−0.671 − 0.387i)12-s + (0.864 − 0.628i)13-s + (3.14 − 1.44i)14-s + (−0.345 + 1.06i)15-s + (−3.06 + 1.36i)16-s + (0.576 − 5.48i)17-s + ⋯
L(s)  = 1  + (−0.921 + 0.0968i)2-s + (1.17 + 1.05i)3-s + (−0.138 + 0.0295i)4-s + (0.0744 + 0.167i)5-s + (−1.18 − 0.858i)6-s + (−0.947 + 0.320i)7-s + (1.00 − 0.326i)8-s + (0.155 + 1.47i)9-s + (−0.0847 − 0.146i)10-s + (0.984 − 0.177i)11-s + (−0.193 − 0.111i)12-s + (0.239 − 0.174i)13-s + (0.841 − 0.386i)14-s + (−0.0891 + 0.274i)15-s + (−0.765 + 0.340i)16-s + (0.139 − 1.33i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.380 - 0.924i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.380 - 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(77\)    =    \(7 \cdot 11\)
Sign: $0.380 - 0.924i$
Analytic conductor: \(0.614848\)
Root analytic conductor: \(0.784122\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{77} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 77,\ (\ :1/2),\ 0.380 - 0.924i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.634300 + 0.425042i\)
\(L(\frac12)\) \(\approx\) \(0.634300 + 0.425042i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (2.50 - 0.847i)T \)
11 \( 1 + (-3.26 + 0.590i)T \)
good2 \( 1 + (1.30 - 0.136i)T + (1.95 - 0.415i)T^{2} \)
3 \( 1 + (-2.02 - 1.82i)T + (0.313 + 2.98i)T^{2} \)
5 \( 1 + (-0.166 - 0.373i)T + (-3.34 + 3.71i)T^{2} \)
13 \( 1 + (-0.864 + 0.628i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-0.576 + 5.48i)T + (-16.6 - 3.53i)T^{2} \)
19 \( 1 + (4.47 + 0.951i)T + (17.3 + 7.72i)T^{2} \)
23 \( 1 + (-1.37 + 2.38i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.06 - 0.345i)T + (23.4 + 17.0i)T^{2} \)
31 \( 1 + (2.75 - 6.19i)T + (-20.7 - 23.0i)T^{2} \)
37 \( 1 + (4.49 + 4.98i)T + (-3.86 + 36.7i)T^{2} \)
41 \( 1 + (-2.68 - 8.25i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 4.85iT - 43T^{2} \)
47 \( 1 + (0.543 - 2.55i)T + (-42.9 - 19.1i)T^{2} \)
53 \( 1 + (12.2 + 5.46i)T + (35.4 + 39.3i)T^{2} \)
59 \( 1 + (0.435 + 2.04i)T + (-53.8 + 23.9i)T^{2} \)
61 \( 1 + (6.52 - 2.90i)T + (40.8 - 45.3i)T^{2} \)
67 \( 1 + (-4.12 - 7.14i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-2.08 - 1.51i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (9.49 - 2.01i)T + (66.6 - 29.6i)T^{2} \)
79 \( 1 + (-2.06 + 0.216i)T + (77.2 - 16.4i)T^{2} \)
83 \( 1 + (-2.94 - 2.14i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + (-2.68 - 1.54i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (7.81 + 10.7i)T + (-29.9 + 92.2i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.68232897359385981465831267261, −13.93781831284292666214029413902, −12.70712836849013812014167208508, −10.82127078937091647455426082120, −9.765871695319519583425907430896, −9.130955495223839229253944009262, −8.406965209599221549111441260255, −6.78382026346244788826637026092, −4.49305777751897913524761340342, −3.08534918918524077249302156937, 1.64216262196223965143224546463, 3.78085505659539078681908827795, 6.47383666569848775898362221573, 7.60942327288435874159300298346, 8.733012438515680511818774998878, 9.340419378016792309164689378775, 10.63494010911726838943619735086, 12.47124203179873414525848772942, 13.24731200734568900036055375804, 14.08295078735653357299130488768

Graph of the $Z$-function along the critical line